Theorems · Definition · category theory

CategoryTheory.GrothendieckTopology.sheafification

{C : Type u} →
  [inst : CategoryTheory.Category.{v, u} C] →
    (J : CategoryTheory.GrothendieckTopology C) →
      (D : Type w) →
        [inst_1 : CategoryTheory.Category.{w', w} D] →
          [∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X),
                CategoryTheory.Limits.HasMultiequalizer (S.index P)] →
            [∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D] →
              CategoryTheory.Functor (CategoryTheory.Functor Cᵒᵖ D) (CategoryTheory.Functor Cᵒᵖ D)

The sheafification of a presheaf P, as a functor. NOTE: Additional hypotheses are needed to obtain a proof that this is a sheaf!

Defined in

Mathlib.CategoryTheory.Sites.ConcreteSheafification

Cited by

7 results in Mathlib

Foundations

Depth 53 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasMultiequalizerCategoryTheory.Limits.HasColimitsOfShape